A key benefit of the Matérn covariance in Gaussian process models is that it is possible precisely control the degree of differentiability of the process realizations via a single parameter of the covariance function. However, the Matérn class possesses exponentially decaying tails, and thus may not be suitable for modeling polynomially decaying dependence. This problem can be remedied using polynomial covariances; however one loses control over the degree of differentiability of the process realizations, in that the realizations using polynomial covariances are either infinitely differentiable or not differentiable at all. We construct a new family of covariance functions called the Confluent Hypergeometric (CH) class using a scale mixture representation of the Matérn class where one obtains the benefits of both Matérn and polynomial covariances. The resultant covariance contains two parameters: one controls the degree of differentiability near the origin and the other controls the polynomial tail heaviness, independently of each other. Application using NASA's OCO-2 satellite data confirms the advantage of the CH class over the Matérn class, especially in extrapolative settings.